To solve this problem, we will use the concept of trigonometry, particularly the tangent function in right triangles, and make use of the angles of elevation given from points A and B. Here's the process:

Step 1: Define the Variables

• Let $h$ be the height of the mountain in miles.
• Let $d$ be the horizontal distance from point A to the base of the mountain.
• The distance between A and B is given as $77$ miles.
• The angle of elevation at point A is $13.7^\circ$.
• The angle of elevation at point B is $10.4^\circ$.

From these variables, we can form two right triangles, one at point A and one at point B.

Step 2: Use the Tangent Function

From point A, the tangent of the angle of elevation $\theta_A = 13.7^\circ$ is:

$\tan(13.7^\circ) = \frac{h}{d}$

From point B, which is $77$ miles behind A, the angle of elevation $\theta_B = 10.4^\circ$ gives:

$\tan(10.4^\circ) = \frac{h}{d + 77}$

Step 3: Set up the System of Equations

We now have two equations:

1. $h = d \cdot \tan(13.7^\circ)$
2. $h = (d + 77) \cdot \tan(10.4^\circ)$

Step 4: Solve the System of Equations

Substitute $h = d \cdot \tan(13.7^\circ)$ from the first equation into the second equation:

$d \cdot \tan(13.7^\circ) = (d + 77) \cdot \tan(10.4^\circ)$

Now, we solve for $d$. After that, we can use the value of $d$ to find $h$.

Let me compute that for you.The horizontal distance from point A to the base of the mountain is approximately $234.6$ miles. The height of the mountain is approximately $301,959$ feet.

Do you have any questions or would you like further details?

Here are 5 related questions to deepen your understanding:

1. How does the tangent function relate to the angle of elevation in trigonometry?
2. What is the significance of using different points (A and B) with different angles of elevation?
3. How would the solution change if the distance between A and B was increased or decreased?
4. What assumptions do we make about the mountain's shape in this type of problem?
5. How can we apply the same principles to find the height of buildings or other structures?

Tip: Always check if the angles of elevation are given in degrees or radians, as this affects the computation.